Author : Jean-P. Serre
Publisher : Springer Science & Business Media
Page : 228 pages
File Size : 30,18 MB
Release : 2013-06-29
Category : Technology & Engineering
ISBN : 3663106322
The book is based on a course given by J.-P. Serre at the Collège de France in 1980 and 1981. Basic techniques in Diophantine geometry are covered, such as heights, the Mordell-Weil theorem, Siegel's and Baker's theorems, Hilbert's irreducibility theorem, and the large sieve. Included are applications to, for example, Mordell's conjecture, the construction of Galois extensions, and the classical class number 1 problem. Comprehensive bibliographical references.
Author : Jean Pierre Serre
Publisher : Vieweg+Teubner Verlag
Page : 220 pages
File Size : 38,46 MB
Release : 1989-01-01
Category : Mathematics
ISBN : 9783528089689
Author : John William Scott Cassels
Publisher : Cambridge University Press
Page : 148 pages
File Size : 34,8 MB
Release : 1991-11-21
Category : Mathematics
ISBN : 9780521425308
A self-contained introductory text for beginning graduate students that is contemporary in approach without ignoring historical matters.
Author : Joseph H. Silverman
Publisher : Springer Science & Business Media
Page : 292 pages
File Size : 42,46 MB
Release : 2013-04-17
Category : Mathematics
ISBN : 1475742525
The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book’s accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.
Author : Daniel Huybrechts
Publisher : Cambridge University Press
Page : 499 pages
File Size : 15,52 MB
Release : 2016-09-26
Category : Mathematics
ISBN : 1316797252
K3 surfaces are central objects in modern algebraic geometry. This book examines this important class of Calabi–Yau manifolds from various perspectives in eighteen self-contained chapters. It starts with the basics and guides the reader to recent breakthroughs, such as the proof of the Tate conjecture for K3 surfaces and structural results on Chow groups. Powerful general techniques are introduced to study the many facets of K3 surfaces, including arithmetic, homological, and differential geometric aspects. In this context, the book covers Hodge structures, moduli spaces, periods, derived categories, birational techniques, Chow rings, and deformation theory. Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular, and each chapter ends with questions and open problems. Based on lectures at the advanced graduate level, this book is suitable for courses and as a reference for researchers.
Author : Paulo Ribenboim
Publisher : Springer Science & Business Media
Page : 306 pages
File Size : 48,48 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 1468493426
Lecture I The Early History of Fermat's Last Theorem.- 1 The Problem.- 2 Early Attempts.- 3 Kummer's Monumental Theorem.- 4 Regular Primes.- 5 Kummer's Work on Irregular Prime Exponents.- 6 Other Relevant Results.- 7 The Golden Medal and the Wolfskehl Prize.- Lecture II Recent Results.- 1 Stating the Results.- 2 Explanations.- Lecture III B.K. = Before Kummer.- 1 The Pythagorean Equation.- 2 The Biquadratic Equation.- 3 The Cubic Equation.- 4 The Quintic Equation.- 5 Fermat's Equation of Degree Seven.- Lecture IV The Naïve Approach.- 1 The Relations of Barlow and Abel.- 2 Sophie Germain.- 3 Co.
Author : Henri Darmon
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 39,87 MB
Release : 2004
Category : Mathematics
ISBN : 0821828681
The book surveys some recent developments in the arithmetic of modular elliptic curves. It places a special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplication, Heegner points, and some conjectural variants. The first three chapters introduce the background and prerequisites: elliptic curves, modular forms and the Shimura-Taniyama-Weil conjecture, complex multiplication and the Heegner point construction. The next three chapters introduce variants of modular parametrizations in which modular curves are replaced by Shimura curves attached to certain indefinite quaternion algebras. The main new contributions are found in Chapters 7-9, which survey the author's attempts to extend the theory of Heegner points and complex multiplication to situations where the base field is not a CM field. Chapter 10 explains the proof of Kolyvagin's theorem, which relates Heegner points to the arithmetic of elliptic curves and leads to the best evidence so far for the Birch and Swinnerton-Dyer conjecture.
Author : Joseph H. Silverman
Publisher : Springer Science & Business Media
Page : 414 pages
File Size : 20,46 MB
Release : 2013-03-09
Category : Mathematics
ISBN : 1475719205
The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.
Author : W. W. L. Chen
Publisher : Cambridge University Press
Page : 493 pages
File Size : 43,50 MB
Release : 2009-02-19
Category : Mathematics
ISBN : 0521515386
A collection of papers inspired by the work of Britain's first Fields Medallist, Klaus Roth.
Author : H. Kisilevsky
Publisher : American Mathematical Soc.
Page : 329 pages
File Size : 36,37 MB
Release : 2004
Category : Mathematics
ISBN : 0821833316
Collects articles from the meeting of the Canadian Number Theory Association held at the Centre de Recherches Mathematiques (CRM) at the University of Montreal. This book covers topics such as algebraic number theory, analytic number theory, arithmetic algebraic geometry, computational number theory, and Diophantine analysis and approximation.
